Reliability analysis method for functionally graded plates based on adaptive kriging model

Abstract

This invention discloses a reliability analysis method for functionally graded plates (FGPs) based on an adaptive Kriging model, relating to the field of mechanical engineering, to address the problems of low computational efficiency and poor accuracy. The method includes: establishing a finite element model and limit state function of the FGP; using uncertain parameters as evidence variables, establishing an identification framework and power set for independent and related evidence variables; using the Cartesian product method to establish a joint identification framework including all evidence variables; using Latin hypercube sampling to select sample points and obtaining the corresponding limit state function values; establishing a Kriging model based on the sample points and limit state function values ​​using a zero-order polynomial and a Gaussian function; updating the Kriging model based on the maximum false positive probability criterion until convergence is met; calculating the extreme values ​​of all joint focal elements using the updated Kriging model, and obtaining a reliability index based on the basic probability allocation of the joint focal elements, thereby improving computational efficiency and accuracy.

Description

Technical Field

[0001] This invention belongs to the field of mechanical engineering, specifically relating to a reliability analysis method for functionally graded plates based on an adaptive Kriging model. Background Technology

[0002] Most existing research on functionally graded plates (FRPs) focuses on deterministic models. However, in actual service, the thermodynamic response of FRPs exhibits uncertainty due to factors such as manufacturing limitations, measurement errors, insufficient understanding of thermal issues, and the inherent uncertainties of external loads. Ignoring these uncertainties during structural design without conducting reliability analysis could lead to catastrophic consequences.

[0003] Reliability analysis methods based on stochastic methods are relatively mature, but they require a large amount of sample data to establish the probability density function of uncertain parameters, which limits the further application of stochastic methods in reliability analysis on functionally graded boards. In contrast, reliability analysis methods based on evidence theory can strictly establish the evidence variables and basic probability assignments of uncertain parameters according to the amount of data obtained, and conduct reliability analysis based on this, effectively avoiding the problem of insufficient experimental data. Most existing reliability analysis methods based on evidence theory assume that uncertain parameters are uncorrelated, but in actual engineering, different uncertain parameters often have some correlation. Ignoring the correlation between parameters can lead to an overly conservative analysis structure.

[0004] In reliability analysis models, the reliability and failure domains in the uncertainty space are typically determined by identifying the limit state function between the input variables and the output response, and then the reliability index of the structure is calculated. Due to the complexity of real-world engineering projects, the explicit limit state function between the input variables and the output response is often unavailable. Approximation methods such as Monte Carlo simulations require extensive calls to simulation programs to calculate the limit state function values ​​at sample points, resulting in unacceptable computational costs. Therefore, establishing a surrogate model to replace the true limit state function through reasonable experimental design, and using this surrogate model for reliability analysis, has become a method to reduce computational costs and improve computational efficiency. The Kriging model, through reasonable experimental design, collects input variables and output responses and establishes an explicit function to replace the limit state function. Its advantages, such as applicability to high-dimensional functions, good fitting to nonlinear functions, and the ability to provide prediction variance, have led to its widespread application in structural reliability analysis.

[0005] However, there is still a lack of research on Kriging models for reliability analysis under the evidence theory model. It is necessary to establish a suitable evidence theory model to characterize the correlation between different uncertain parameters, and on this basis, to establish an efficient and high-precision Kriging model for reliability analysis of functionally graded plates. Summary of the Invention

[0006] In view of this, the present invention provides a reliability analysis method for functional graded plates based on an adaptive kriging model, which solves the problems of low computational efficiency and poor accuracy caused by ignoring the correlation of uncertain parameters in reliability analysis of functional graded plates.

[0007] This invention provides a method for reliability analysis of functionally graded plates based on an adaptive kriging model, comprising:

[0008] Step 1: Discretize the geometric model of the functionally graded plate using a finite element mesh to obtain the finite element model of the functionally graded plate. Perform thermo-mechanical coupling analysis on the finite element model of the functionally graded plate to obtain the thermo-mechanical response of the functionally graded plate. Based on the thermo-mechanical response and a given threshold, establish a limit state function characterizing the reliability of the functionally graded plate.

[0009] Step 2: The uncertain parameters of the functional gradient plate are used as evidence variables, and the evidence variables are divided into multiple variable groups. An identification framework and power set are established for each variable group. Each variable group has at least one evidence variable. Evidence variables in different variable groups are independent, while evidence variables in the same variable group are related. When a variable group has one evidence variable, an interval is used to establish the corresponding identification framework. When a variable group has at least two evidence variables, an ellipsoid is used to establish the corresponding identification framework.

[0010] Step 3: Based on the variable set and the corresponding power set, use the Cartesian product method to establish the joint focal element and the corresponding basic probability assignment of all the evidence variables;

[0011] Step 4: Determine a hypercube that includes the range of values ​​for all the evidence variables, select sample points within the hypercube using Latin hypercube sampling, and obtain the limit state function values ​​at the sample points through finite element analysis.

[0012] Step 5: Establish the Kriging model for the sample points and the corresponding limit state function values, where the zeroth-order polynomial is the basis function of the Kriging model and the Gaussian function is the correlation function of the stochastic process term;

[0013] Step 6: Calculate the extreme value misclassification probability of the Kriging model on all the joint focal elements, obtain the extreme point corresponding to the maximum extreme value misclassification probability, and obtain the limit state function value at the extreme point through finite element analysis. Use the extreme point and the corresponding limit state function value as new data to update the Kriging model until the convergence condition is met.

[0014] Step 7: Obtain the extreme values ​​of the updated Kriging model in all joint focal elements, and determine the reliability index of the functional gradient plate based on each extreme value and the corresponding basic probability assignment.

[0015] In some possible embodiments, the extreme value misclassification probability of the Kriging model at all the joint focal elements is calculated to obtain the extreme point corresponding to the maximum extreme value misclassification probability, and the limit state function value at the extreme point is obtained through finite element analysis. The extreme point and the corresponding limit state function value are used as new data to update the Kriging model until the convergence condition is met, including:

[0016] When the reliability index is confidence, the extreme point of any of the joint focal elements is the point within the joint focal element where the probability of the limit state function value being less than 0 is determined by the Kriging model. Based on the extreme point of the joint focal element and the corresponding basic probability allocation, the maximum absolute error between the expected and true values ​​of the reliability index is calculated.

[0017] When the reliability index is similarity, the extreme point of any of the joint focal elements is the point within the joint focal element where the probability of the limit state function value being greater than 0 is determined by the Kriging model. Based on the extreme point of the joint focal element and the corresponding basic probability allocation, the maximum absolute error between the expected value and the true value of the reliability index is calculated.

[0018] Based on the maximum absolute error, the extreme point corresponding to the maximum probability of extreme value misjudgment is determined, and the Kriging model is updated based on the extreme point and the corresponding limit state function value until the maximum absolute error between the expected reliability index obtained from the updated Kriging model and the true value is less than a threshold.

[0019] The functional graded plate reliability analysis method based on an adaptive kriging model provided by this invention has at least the following advantages:

[0020] (1) Compared with traditional reliability analysis models, the Kriging model established in this embodiment of the invention fully considers the correlation of uncertain parameters in actual engineering. It uses intervals and ellipsoids to establish an identification framework for independent evidence variables and related evidence variables. The reliability index calculation results are more accurate than those of methods that do not consider correlation.

[0021] (2) This invention uses a Kriging model to replace the actual limit state function for reliability analysis, which avoids calling a lot of time-consuming simulation models when solving reliability indexes and improves computational efficiency.

[0022] (3) Based on the calculation characteristics of the reliability index under the evidence theory model, the Kriging model is updated efficiently by adding points at the joint focal element extreme points, and the learning process is terminated in a timely manner by controlling the absolute error of the reliability index. While ensuring the accuracy of the reliability analysis, the number of sample points required to construct the Kriging model is reduced, overcoming the problem that existing technologies require too many sample points to construct the Kriging model. Attached Figure Description

[0023] Figure 1 This is a flowchart of a functional gradient plate reliability analysis method based on an adaptive Kriging model according to an embodiment of the present invention;

[0024] Figure 2 This is a simplified flowchart of a functional graded plate reliability analysis method based on an adaptive Kriging model according to an embodiment of the present invention.

[0025] Figure 3 This is a schematic diagram of a functional gradient plate according to an embodiment of the present invention;

[0026] Figure 4 This is a front view of the functional gradient plate according to an embodiment of the present invention;

[0027] Figure 5 This is a side view of a functional gradient plate according to an embodiment of the present invention.

[0028] Explanation of reference numerals in the attached figures:

[0029] 10-Top panel;

[0030] 20-Web;

[0031] 30 - Bottom panel. Detailed Implementation

[0032] To make the above-mentioned objectives, features, and advantages of the embodiments of the present invention more apparent and understandable, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] See Figure 1 and Figure 2 This invention provides a reliability analysis method for functionally graded plates based on an adaptive Kriging model. The method specifically includes the following steps:

[0034] Step 1: Discretize the geometric model of the functionally graded plate using a finite element mesh to obtain the finite element model of the functionally graded plate. Perform thermo-mechanical coupling analysis on the finite element model of the functionally graded plate to obtain the thermo-mechanical response of the functionally graded plate. Based on the thermo-mechanical response and a given threshold, establish the limit state function characterizing the reliability of the functionally graded plate.

[0035] Specifically, the geometric model of the functionally graded plate is discretized using a finite element mesh to obtain its finite element model. This finite element model is then used for thermo-mechanical coupling analysis to obtain the thermo-mechanical response S, which characterizes the reliability of the functionally graded plate. Based on the thermo-mechanical response S and a given threshold G0, the limit state function G = G0 - S, characterizing the reliability of the functionally graded plate, can be established.

[0036] For example, the geometric model of a functional gradient plate is as follows: Figure 3 As shown, the geometric dimensions are as follows Figure 4 and Figure 5 As shown. In some possible embodiments, the geometrical parameters of the functionally graded plate are shown in Table 1:

[0037] Table 1 Geometric Dimensions of Functionally Graded Plates

[0038] 2.3 3.1 8.06 150 142.8 75 107 90

[0039] The functionally graded plate includes an upper panel 10, a web 20, and a lower panel 30, wherein the web 20 has openings. The upper panel 10 is made of ceramic composite material, the lower panel 30 is made of carbon fiber reinforced phenolic resin, and the web 20 is made of a ceramic-phenolic resin gradient transition material with a power-law transition form, as shown below:

[0040]

[0041] Where V r (z) represents the volume fraction of phenolic resin at position z, V c (z) represents the volume fraction of the ceramic at point z, p is the power exponent characterizing the transition form, and h is the height of the web 20. The macroscopic physical properties of the web 20 at point z are:

[0042]

[0043] Among them, P e (z) represents the macroscopic physical property parameter of the web 20 at point z, P r P c These represent the physical properties of phenolic resin and ceramics, including elastic modulus, Poisson's ratio, specific heat, thermal conductivity, coefficient of thermal expansion, and density.

[0044] The geometric model of the functionally graded plate was discretized using 6688 hexahedral elements. Simply supported boundary conditions were applied to the four sides of the upper panel 10 and the lower panel 30 of the functionally graded plate, and a uniformly distributed heat flow was applied to the top of the upper panel 10. The data on the change of the applied heat flow over time are shown in Table 2.

[0045] Table 2 Data on heat flux variation over time

[0046]

[0047] Using a finite element model of a functionally graded plate (FRP), the maximum von Mises stress of the FRP over 6000 s is calculated. This maximum von Mises stress is then used as the thermodynamic response S, a limit state function characterizing the reliability of the FRP. Based on this thermodynamic response S and a given threshold G0 = 3.9 × 10⁻⁶, the maximum von Mises stress is calculated. 2 MPa, establish the limit state function G characterizing the reliability of the functionally graded plate. Specifically, the limit state function G is: G = 3.9 × 10 2 MPa-S. For an uncertain parameter X, G(X) > 0 indicates that the functionally graded plate is safe, and G(X) ≤ 0 indicates that the functionally graded plate is unsafe.

[0048] Step 2: Use the uncertain parameters of the functional gradient plate as evidence variables, and divide the evidence variables into multiple variable groups. Establish the identification framework and power set for each variable group. Each variable group has at least one evidence variable. Evidence variables in different variable groups are independent, while evidence variables in the same variable group are related. When a variable group has one evidence variable, an interval is used to establish the corresponding identification framework. When a variable group has at least two evidence variables, an ellipsoid is used to establish the corresponding identification framework.

[0049] Specifically, a functional gradient plate may include k uncertain parameters, and these k uncertain parameters are all used as evidence variables X = (X1, X2, ..., X...). k ), where X m Let m = 1, 2, ..., k represent the m-th evidentiary variable. The evidentiary variables are divided into multiple variable groups, each group containing at least one evidentiary variable; that is, the evidentiary variables are divided into r groups, each group containing k... i There are i = 1, 2, 3, ..., r evidential variables. Among them, the evidential variables in different variable groups are independent of each other, and the evidential variables in the same variable group are correlated with each other.

[0050] Understandably, when there is only one evidential variable in a variable group, that evidential variable is independent of the other evidential variables and is considered an independent evidential variable. When there are two or more evidential variables in a variable group, these evidential variables are correlated with each other and are independent of the other evidential variables, forming a set of correlated evidential variables.

[0051] For r sets of evidence variables, the identification framework Ω1,Ω2,…,Ω for each set of evidence variables is defined. r , where Ω i ,i=1,2,...,r represents the i-th identification frame, meaning each variable group corresponds to one identification frame. When the i-th group of evidence variables contains relevant evidence variables, n is used. i Establish a representation of k using ellipsoids. i Relevant evidence variables Identification framework for the range of values ​​Ω i :

[0052] Ω i ={A i (1),A i (2),...,A i (n i )}

[0053] Where A i (l)l=1,2,...n i Represents the recognition frame Ω i The included l-th ellipsoid, A i The expression for (l) is:

[0054]

[0055] in, and W i,l Let be the midpoint vector and characteristic matrix of the l-th ellipsoid, respectively.

[0056] When the j-th group of evidence variables are independent evidence variables, the j-th group of evidence variables contains only one evidence variable X. j,1 , using n j Establish a representation of X in each interval. j,1 Identification framework for the range of values ​​Ω j :

[0057] Ω j ={A j (1),A j (2),...,A j (n j )}

[0058] Among them, A j (l)l=1,2,...n j Represents the recognition frame Ω j The l-th interval contained therein, A j The expression for (l) is:

[0059]

[0060] in, These are the lower and upper bounds of the l-th interval, respectively.

[0061] According to each identification frame Ω i The corresponding power set Θ is obtained. i i = 1, 2, ..., r, which is represented as:

[0062] Θ i ={{A i (1)},{A i (2)},...,{A i (n i )},{A i (1),A i (2)},{A i (1),A i (3)},…,Ω i}

[0063] Power set Θ i The included ellipsoid or interval {A i (l)}, i=1,2,…r,l=1,2,n i Let A be the focal element. Since the focal element {A} i (l)} contains only one element A i (l) will still focus on the element {A} i (l)} is denoted as A i (l). Jiao Yuan A i The basic probability distribution of (l) is m(A) i (l)), i = 1, 2, ..., r, l = 1, 2, n i .

[0064] For example, the uncertain parameters of a functionally graded plate include the thermal conductivity K of the phenolic resin material. m and specific heat capacity C m The thermal conductivity K of ceramics c and specific heat capacity C c And the power exponent p, which characterizes the material composition distribution of web 20, are also included as evidence variables. Of these five evidence variables, the power exponent p is independent of the other four. The thermal conductivity K of the phenolic resin material is also considered. m and specific heat capacity C m Relatedly, the thermal conductivity K of ceramics c and specific heat capacity C c Related, the material properties of different types of materials are independent of each other.

[0065] Based on the correlations among the five evidentiary variables, the evidentiary variables are classified into three groups {{K} m C m},{K c C c},{p}}, where the thermal conductivity K of the phenolic resin material is... m and specific heat capacity C m As a set of relevant evidence variables, the thermal conductivity K of ceramics c and specific heat capacity C c Let Ω1, Ω2, and Ω3 be a set of relevant evidence variables, and p be an independent evidence variable. An identification framework of three sets of evidence variables, Ω1, Ω2, and Ω3, is established using intervals or ellipsoids. The elements within Ω1 represent the thermal conductivity K of the aldehyde resin. m and specific heat capacity C m The ellipsoid of the uncertainty domain; the elements within Ω² represent the thermal conductivity K of the ceramic material. c and specific heat capacity C c The ellipsoid of the uncertainty region, Ω3, contains elements representing the intervals of the uncertainty region of the power exponent p. The power sets corresponding to the identification frames Ω1, Ω2, and Ω3 are denoted as Θ1, Θ2, and Θ3, respectively. The information of the focal elements contained in the power sets Θ1, Θ2, and Θ3 are shown in Tables 3, 4, and 5, respectively.

[0066] Table 3 contains information about the focal element in the power set Θ1.

[0067]

[0068] Table 4 contains information about the focal element in the power set Θ2.

[0069]

[0070] Table 5 contains information about the focal element in the power set Θ3.

[0071]

[0072] Step 3: Based on the variable set and its corresponding power set, use the Cartesian product method to establish the joint focal element of all evidence variables and the corresponding basic probability assignment.

[0073] Specifically, based on multiple sets of evidence variables and their corresponding power sets, a joint identification framework and joint focal element for all evidence variables are established using the Cartesian product method, as follows:

[0074] Ω=Θ1×Θ2×…×Θ r =

[0075] {A i =[A1(l1),A2(l2),…,A r (l r )],A1(l1)∈Θ1,A2(l2)∈Θ2,…,A r (l r )∈Θ r}

[0076] Where Ω represents the joint identification frame, and "×" denotes the Cartesian product operation. Ω contains element A. i For the i-th, i = 1, 2, ..., n1×n2×…×n of the joint identification framework r A joint focal element, A i (l i ) for identifying the Ω i The lth i Using the direct multiplication method, the basic confidence assignment of the joint focal elements can be obtained as follows:

[0077]

[0078] Based on the above example, that is... Figure 3 Based on the functional gradient board shown as an example, the joint identification framework for all evidence variables is as follows:

[0079] Ω=Θ1×Θ2×Θ3=

[0080] {A i =[A1(l1),A2(l2),A3(l3)],A1(l1)∈Θ1,A2(l2)∈Θ2,A3(l3)∈Θ3}

[0081] The basic confidence level allocation for the joint coulomb is as follows:

[0082]

[0083] Step 4: Determine a hypercube that includes the range of all evidence variables. Select sample points within the hypercube using Latin hypercube sampling. Obtain the limit state function values ​​of the sample points through finite element analysis.

[0084] First, based on the recognition framework Ω1,Ω2,…,Ω from step two. r Calculate the upper and lower bounds of the range of values ​​for the evidence variables. For relevant evidence variables... Corresponding recognition framework Ω i First, the covariance matrix of each ellipsoid is obtained through matrix inversion. The recognition frame Ω is then determined by the following formula. i Evidence variables of the l-th ellipsoid included Upper and lower bounds:

[0085]

[0086]

[0087] in, and Evidence variables At focal element A on the ellipsoid i The vector formed by the lower and upper bounds of (l). Calculate After considering the upper and lower bounds of all ellipsoids, a vector formed by comparing the lower and upper bounds can be obtained. and

[0088] The identification framework Ω corresponding to independent evidence variables j It contains only one evidence variable X. j,1 It can be identified by the Ω frame. j The upper and lower bounds of the focal element within the included interval are indeed used to obtain the evidence variable X. j,1 The lower and upper bounds and This leads to a hypercube space containing the range of values ​​for all evidence variables:

[0089] Using Latin hypercube sampling, n sample points are selected within the hypercube space to establish a sample point set D = [X1, X2, ..., X...]. n ], where X j =(X 1,j ,X 2,j ,...,X k,j ) T (i = 1, 2, ..., n) A Let X be a sample vector containing k evidential variables. i,j Let represent the j-th value of the i-th evidence variable. Through finite element analysis, the set of limit state function responses at n different sample points is obtained.

[0090] For example, the thermal conductivity K of the phenolic resin material in the functionally graded plate m and specific heat capacity C m The thermal conductivity K of ceramics c and specific heat capacity C c And with the power exponent p being an uncertain parameter, the hypercube space encompassing the range of all evidence variables can be represented as:

[0091]

[0092] Within the aforementioned hypercube space, 28 sample points are selected using Latin hypercube sampling to establish a sample point set D = [X1, X2, ..., X...]. 28 The limit state function response set G = [G(X1), G(X2), ..., G(X)] can be obtained at the sample points through finite element analysis. 28 )).

[0093] Step 5: Establish a Kriging model of the sample points and their corresponding limit state function values, where the zeroth-order polynomial is the basis function of the Kriging model and the Gaussian function is the correlation function of the stochastic process term.

[0094] Specifically, for a sample set D = [X1, X2, ..., Xn] consisting of n sample points... n ], where the dimension of the sample is k, and through finite element analysis, the limit state function values ​​G=[G(X1),G(X2),…,G(X) are obtained at n sample points. n The Kriging model established from the sample set and the limit state function values ​​can be expressed as:

[0095] G(X)=f T (X)β+ε(X)

[0096] Where f(X)=[f1(X),f2(X),...,f q (X)] T Let f be the basis function vector of the Kriging model, using zeroth-order polynomials as basis functions. In this case, the regression term is a constant, i.e., f T (X)β=β.

[0097] The stochastic process term z(X) represents a steady-state Gaussian process, and z(X) ~ N(0,σ). 2 The covariance of the error term between any two points X and Y is:

[0098] Cov[z(X),z(Y)]=σ 2 R θ (X,Y)

[0099] Where, σ 2 R represents the process variance. θ (X,Y) represents the correlation function between two points; here, the Gaussian correlation function is used.

[0100]

[0101] Among them, X j and Y j Let θ represent the j-th uncertain parameter value of X and Y, respectively. j The component of the correlation coefficient θ on the j-th uncertain parameter can be obtained through maximum likelihood estimation:

[0102]

[0103] R represents a symmetric correlation coefficient matrix, derived from the sample set D = [X1, X2, ..., X...]. n The elements of matrix R are:

[0104] R ij=R(X) i ,X j ), i,j=1,2,...,n

[0105] Regression coefficient vector β and process variance σ 2 The estimated value can be expressed as:

[0106]

[0107]

[0108] Here, 1 represents a k×1 vector whose elements are all 1s.

[0109] The function value of any point X in the Kriging model follows a normal distribution:

[0110]

[0111] in:

[0112]

[0113]

[0114] in:

[0115] r(X)=[R(X1,X),R(X2,X),...,R(X n ,X)]

[0116] u(X) = 1 T Rr(X)-f(X)

[0117] For example, the thermal conductivity K of the phenolic resin material in step four m and specific heat capacity C m The thermal conductivity K of ceramics c and specific heat capacity C c Based on the example of the power exponent p-value, using a zero-order polynomial function as the basis function and a Gaussian function as the correlation function of the stochastic process term, maximum likelihood estimation yields θ = (1.7678, 0.3397, 0.1, 0.1, 0.1061). σ is then obtained through matrix operations. 2 =7.8782.

[0118] Step 6: Calculate the extreme value misclassification probability of the Kriging model on all joint focal elements, obtain the extreme point corresponding to the maximum extreme value misclassification probability, and obtain the limit state function value at the extreme point through finite element analysis. Use the extreme point and the corresponding limit state function value as new data to update the Kriging model until the convergence condition is met.

[0119] Specifically, n is generated within the aforementioned hypercube space encompassing the range of all evidence variables using Latin hypercube sampling. candidate For each candidate point, its coordinates are evaluated to determine which point falls within the joint focal element A. i i = 1, 2, ..., n A Candidate point set within i = 1, 2, ..., n A .

[0120] A kriging model is established based on the sample points and their corresponding limit state function values. After the kriging model is established, when the reliability index is defined as Bel or Pl, let there be n under the joint identification framework Θ. A There are _ ... i i = 1, 2, ..., n A The extreme points within are defined as:

[0121]

[0122] in, and These represent the joint focal element A when calculated based on Bel and Pl, respectively. i i = 1, 2, ..., n A The extreme point, i = 1, 2, ..., n A Let X be the set formed by these elements. Bel , i = 1, 2, ..., n A Let X be the set formed by these elements. Pl Therefore, the probability of misjudging the sign of the extreme value of each joint focal element can be calculated using the following formula:

[0123]

[0124] Where Φ(X) represents the cumulative probability density function of the standard normal distribution. According to the definition of reliability indices under evidence theory, credibility Bel and plausibility Pl are defined as follows:

[0125]

[0126] Among them, I Bel (A i ) and I Pl (A i All of these are indicator functions, defined as follows:

[0127]

[0128] When the Kriging model is used to replace the actual limit state function, the limit state function value G(X) at any point X follows a normal distribution. Therefore, based on the above indicator function, the probability that the extreme value of each joint focal element is greater than 0 is determined to be the probability that the extreme point is greater than 0:

[0129]

[0130] Among them, I Bel (A i ) and I Pl (A i Since Bel and Pl follow a 0-1 distribution, and according to their definitions, Bel and Pl are both random variables following a generalized Poisson binomial distribution, their expected values ​​can be calculated.

[0131]

[0132] The confidence intervals for confidence level and similarity were obtained at 95% using the Bootstrap resampling method.

[0133]

[0134] Furthermore, the maximum absolute error e between the expected and actual reliability index under the Kriging model can be calculated. max :

[0135]

[0136] Given a threshold δ based on the required accuracy of the reliability index, if the maximum absolute error e between the expected and actual values ​​of the reliability index is... max If the value is greater than the threshold δ, compare e. Bel and e Pl Size. If e Bel ≥e Pl The set of extreme points X when calculated based on Bel Bel In the process, find the extreme point that is most easily misjudged; otherwise, in the calculation based on Pl, find the set of extreme points X. Pl Among the points, find the extreme point that is most easily misjudged:

[0137]

[0138] It is understandable that when the reliability index is confidence, the extreme point of any joint focal element is the point within the joint focal element where the probability of the limit state function value being less than 0 is highest, as determined by the Kriging model. Based on the extreme points of the joint focal elements and the corresponding basic probability distribution, the maximum absolute error between the expected and true values ​​of the reliability index is calculated.

[0139] When the reliability index is similarity, the extreme point of any joint focal element is the point within the joint focal element where the probability of the limit state function value being greater than 0, as determined by the Kriging model, is the highest. Based on the extreme points of the joint focal elements and the corresponding basic probability distribution, the maximum absolute error between the expected reliability index and the true value is calculated.

[0140] The limit state function value G(X) at the extreme point is calculated using the finite element method. new ), and the extreme point X new and the limit state function value G(X) at the extreme point new As new data, the Kriging model is updated. Based on the updated Kriging model, the expected reliability indices E(Bel) and E(Pl), as well as the maximum absolute error e between the expected and actual reliability indices, are recalculated. max Updates will stop when the following conditions are met:

[0141] In three consecutive updates of the Kriging model, the maximum absolute error e between the expected and true values ​​of the reliability index is... max Less than the threshold δ, and the calculated value of the reliability index and It remains unchanged. (Among them) and The calculation formula is:

[0142]

[0143] The above formula means that if the combined focal element A i i = 1, 2, ..., n A If the mean minimum of the limiting state function determined by the Kriging model is greater than 0, then the basic probability assignment of the joint focal element is included. If we combine coke element A i i = 1, 2, ..., n A If the mean maximum of the limiting state function determined by the Kriging model is greater than 0, then the basic probability assignment of the joint focal element is included.

[0144] For example, the thermal conductivity K of the phenolic resin material in the functionally graded plate m and specific heat capacity C m The thermal conductivity K of ceramics c and specific heat capacity C c Within the hypercube space formed by the range of values ​​for the power exponent p, 10 values ​​are generated using Latin hypercube sampling within the aforementioned hypercube space containing the ranges of all evidence variables. 5 For each candidate point, its coordinates are evaluated to determine if it falls within each joint focal element A. i The set of candidate points within i = 1, 2, ..., 80 i = 1, 2, ..., 80.

[0145] Based on the Kriging model established in step five, the extreme points of each joint focal element are determined:

[0146]

[0147] Then the Kriging model was calculated in the joint focal element A. i Find the extreme values ​​within the range; and calculate the expected values ​​of Bel and Pl: E(Bel) = 0.2449, E(Pl) = 0.8493. Obtain the 95% confidence intervals for confidence and similarity using the Bootstrap resampling method:

[0148]

[0149] Furthermore, the maximum absolute error e between the expected and actual reliability index under the Kriging model can be calculated. max :

[0150]

[0151] Because of e Bel >e Pl The set of extreme points X when calculated based on Bel Bel Among the points, find the extreme point that is most easily misjudged:

[0152]

[0153] The limit state function value G(X) at the extreme point is calculated using the finite element method. new ), and the extreme point X new and the limit state function value G(X) at the extreme point new As new data, the Kriging model is updated. After adding 189 sample points based on the above Kriging model update method, the constructed Kriging model satisfies the following convergence condition, and the update stops:

[0154] In three consecutive updates of the Kriging model, the maximum absolute error e between the expected and true values ​​of the reliability index is... max All are less than the threshold δ = 0.001, and the calculated value of the reliability index is... and It remains unchanged.

[0155] Step 7: Obtain the extreme values ​​of the updated Kriging model in all joint focal elements, and determine the reliability index of the functional gradient plate based on each extreme value and the corresponding basic probability assignment.

[0156] Specifically, based on the Kriging model that satisfies the maximum absolute error convergence condition between the reliability index and the true value in step six, the confidence level reflecting the upper and lower bounds of the functionally graded plate reliability index is obtained. And realism for:

[0157]

[0158] In summary, the functional gradient plate reliability analysis method based on the adaptive Kriging model provided in this invention establishes an identification framework for independent evidence parameters using intervals and an identification framework for related evidence parameters using ellipsoids, taking into account the correlation between uncertain parameters. On the other hand, the Kriging model, based on the reliability index of the evidence theory model, adds points at the location with the highest probability of misjudgment of the joint focal element extremum to efficiently fit the true limit state function, has the ability to actively learn, reduces the number of calls to the finite element model, and improves computational efficiency while ensuring the accuracy of reliability index calculation.

[0159] The various embodiments or implementation methods described in this specification are presented in a progressive manner. Each embodiment focuses on the differences from other embodiments, and the same or similar parts between the embodiments can be referred to each other.

[0160] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "illustrative embodiment," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with an embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0161] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A functionally graded plate reliability analysis method based on an adaptive Kriging model, characterized by, include: Step 1: Discretize the geometric model of the functionally graded plate using a finite element mesh to obtain the finite element model of the functionally graded plate. Perform thermo-mechanical coupling analysis on the finite element model of the functionally graded plate to obtain the thermo-mechanical response of the functionally graded plate. Based on the thermo-mechanical response and a given threshold, establish a limit state function characterizing the reliability of the functionally graded plate. Step 2: The uncertain parameters of the functional gradient plate are used as evidence variables, and the evidence variables are divided into multiple variable groups. An identification framework and power set are established for each variable group. Each variable group has at least one evidence variable. The evidence variables of different variable groups are independent, and the evidence variables of the same variable group are related. When a variable group has one evidence variable, an interval is used to establish the corresponding identification framework. When a variable group has at least two evidence variables, an ellipsoid is used to establish the corresponding identification framework. Step 3: Based on the variable set and the corresponding power set, use the Cartesian product method to establish the joint focal element and the corresponding basic probability assignment of all the evidence variables; Step 4: Determine a hypercube that includes the range of values ​​for all the evidence variables, select sample points within the hypercube using Latin hypercube sampling, and obtain the limit state function values ​​at the sample points through finite element analysis. Step 5: Establish the Kriging model for the sample points and the corresponding limit state function values, where the zeroth-order polynomial is the basis function of the Kriging model and the Gaussian function is the correlation function of the stochastic process term; Step 6: Calculate the extreme value misclassification probability of the Kriging model on all the joint focal elements, obtain the extreme point corresponding to the maximum extreme value misclassification probability, and obtain the limit state function value at the extreme point through finite element analysis. Use the extreme point and the corresponding limit state function value as new data to update the Kriging model until the convergence condition is met. Step 7: Obtain the extreme values ​​of the updated Kriging model in all joint focal elements, and determine the reliability index of the functional gradient plate based on each extreme value and the corresponding basic probability assignment.

2. The reliability analysis method of functionally graded plates based on the adaptive Kriging model according to claim 1, wherein, Calculate the extremum misclassification probability of the Kriging model on all the joint focal elements, obtain the extremum point corresponding to the maximum extremum misclassification probability, and obtain the limit state function value at the extremum point through finite element analysis. Use the extremum point and the corresponding limit state function value as new data to update the Kriging model until the convergence condition is met, including: When the reliability index is confidence, the extreme point of any of the joint focal elements is the point within the joint focal element where the probability of the limit state function value being less than 0 is determined by the Kriging model. Based on the extreme point of the joint focal element and the corresponding basic probability allocation, the maximum absolute error between the expected and true values ​​of the reliability index is calculated. When the reliability index is similarity, the extreme point of any of the joint focal elements is the point within the joint focal element where the probability of the limit state function value being greater than 0 is determined by the Kriging model. Based on the extreme point of the joint focal element and the corresponding basic probability allocation, the maximum absolute error between the expected value and the true value of the reliability index is calculated. Based on the maximum absolute error, the extreme point corresponding to the maximum probability of extreme value misjudgment is determined, and the Kriging model is updated based on the extreme point and the corresponding limit state function value until the maximum absolute error between the expected reliability index obtained from the updated Kriging model and the true value is less than a threshold.

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